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Let $\triangle ABC$ be a triangle with circumcircle (O) P is on BC, PA is tangent to (O). $E \in PO$, $D \in BE$$AD \bot AB$. Prove that $\angle EAB=\angle ACD$. Please provide an elementary proof for this.

enter image description here and

enter image description here $$\angle CBF =\angle DCA \Leftrightarrow \frac{\sin{\angle BAE}}{\sin{\angle EAO}} =\frac{\sin{\angle ACD}}{\sin{\angle DCH}} =\frac{AD*HC}{DH*AC}$$ $$\frac{\sin{\angle BAE}}{\sin{\angle EAO}} =\frac{AD*CH*\sin{\angle EOB}}{DH*BA*\sin{\angle AOE}}$$ It only need to prove that: $$\frac{AB*HC}{BH*AC} =\frac{\sin{\angle EOB}}{\sin{\angle AOE}} =\frac{\sin{\angle OPB}*\sin{\angle PAB}}{\sin{\angle OPA}*\sin{\angle PBA}}$$ But I have no idea about it. Any ideas?

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  • $\begingroup$ Please show your effort, otherwise your question would be likely put on hold $\endgroup$ – Oldboy Feb 10 '19 at 6:53
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    $\begingroup$ @qsa Math SE is not Brilliant. You can show your work or also show the inspiration to the question (more details here). $\endgroup$ – Toby Mak Feb 10 '19 at 8:21
  • $\begingroup$ The sentence "I tried" is not sufficient. In your attempts, you have probably obtained some results beyond the text of the question, which you haven't yet been able to make use of. Do write them out. $\endgroup$ – Joce Feb 14 '19 at 12:47
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    $\begingroup$ If you're still working on this: you have a property of angles formed by tangents to circumcircle at triangle vertices (see en.wikipedia.org/wiki/Circumscribed_circle#Angles ) $\endgroup$ – Joce Feb 25 '19 at 12:22
  • $\begingroup$ Is this a take home exam or something similar one? $\endgroup$ – C.F.G Feb 19 at 14:54
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two equal angles With the conditions as given, extend $AE$ to $F$, $CD$ to $G$, and join $FG$.

Since$$\angle ACD=\angle AFG$$then$$\angle ACD=\angle EAB$$ IFF $$\angle EAB=\angle AFG$$that is, IFF$$GF\parallel AB$$They appear parallel in the figure, and can be proven parallel in the following three particular cases.

I. Suppose the extreme case where $E$ coincides with $O$, and hence $D$ coincides with $G$.

first special case

Since by the alternate segment theorem$$\angle BAP=\angle ACB$$while $\angle EAB$ is the complement of $\angle BAP$, and $\angle ACD$ is complement of $\angle ACB$ (because $\angle DCB$ in the semicircle is right), therefore$$\angle EAB=\angle ACD$$Further, since $\angle ACD=\angle AFG$, clearly in this case$$GF\parallel AB$$

II. Next let $E$ lie on $AB$.

second special case Since $F$ now coincides with $B$, and $D$ and $G$ coincide with $A$, then$$\angle ACD=\angle EAB=0$$and $GF$ is "parallel" to $AB$ in that it does not intersect it.

III. Finally, if $E$ coincides with $P$, as in the figure below, then $F$, collinear with $AE$ and always on the circle, must coincide with $A$. $D$ lies on $BE$ beyond $E$. And $G$, collinear with $CD$ and always on the circle, must coincide with $B$.

Hence by the alternate segment theorem$$\angle EAB=\angle ACD$$And since they coincide$$AB\parallel FG$$

third case

Thus at least in these three cases, the proposition is true. A general proof seems to require showing that $GF\parallel AB$ always.

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  • $\begingroup$ Under what argument you claim that "and hence $D$ coincides with $G$"? $\endgroup$ – C.F.G Feb 20 at 10:23
  • $\begingroup$ @C.F.G.--$G$ is where $FD$ crosses the circle, and since $D$ is now on the diameter and $\angle BAD$ is right, $D$ is also on the circle, coinciding with $G$. $\endgroup$ – Edward Porcella Feb 20 at 15:57
  • $\begingroup$ @C.F.G.--Correction: $G$ is where $CD$ crosses the circle... $\endgroup$ – Edward Porcella Feb 20 at 18:20
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Here is a brute force solution. (It is not a canonical solution. I tried to give at least a synthetic solution, but failed. The idea was to connect / to parallely project the given configuration of points (and constructed helper points) on the circle $(ABC)$ to points on the line $PA$. Since $PA$ would have been my choice of an idea to solve, it would be not canonical even if it would work...)


We will show an equivalent result. It is important to have the right order of introducing the points, to get their properties in a fluent story. As it is always the case for me, i am trying to mention and more related "coincidences" in the given geometric constellation, even with the risk of having a long(er) presentation, the "bonus results" being in my eyes always important. (The reader may want to extract the solution from the the many ingredients.) The points are constructed, and their properties are claimed in the light yellow framework, arguments are given on white background.

I was trying to use Pascal's Theorem, but failed. (When doing so for a circle with a point $A$ on it, the "line" $AA$ is the tangent in $A$ to the circle.) The problem with this idea is that it is hard to find a configuration of six points like $AA?BC?$ on the circle $(ABC)$ that is indeed useful and uses / encodes essentially the properties of $C$.

Let's go.


A rectangle $ABA'B'$ is given, let $O$ be the intersection of its diagonals $AA'$, $BB'$. We denote by $(O)$ the circumcircle of the rectangle, $(O)=(ABA'B')$. We denote by $(d)$ any direction parallel to the direction of $AB\| A'B'$. Let us denote the point at infinity on $(d)$ by $\infty=\infty_d$. (So a line contains this infinity point, iff it is parallel to $(d)$.)

Let $P$ be a point on the tangent in $A$ to $(O)$, so $PA\perp AO$.

Let $C$ be the second intersection of $PB$ with $(O)$, $C\ne B$.

Let $X$ be a further point on $(O)$.

We consider the point $E=PO\cap BX$. Let $E^*$ be the intersection $PA\cap E\infty$ (of $PA$ with the parallel through $E$ to $(d)$).

We consider the point $D=AB'\cap BX$. Let $D^*$ be the intersection $PA\cap D\infty$ (of $PA$ with the parallel through $D$ to $(d)$).

$AE$ intersects $(O)$ for a second time in a point denoted by $Y$, here $Y\ne A$.

Let $Z$ be the second point of intersection of $E^*X$ with the circle $(O)$.

Mathematics stackexchange, problem 3107126, dan_fulea

Claim: We have $(d)\|YZ\|AB\|EE^*\|DD^*\|B'A'$.

Proof of the claim: Consider the hexagon $AAYZXB$ inscribed in $(O)$. Using Pascal, the points $AA\cap ZX=E^*$, $AY\cap XB=E$, $YZ\cap AB$ are colinear. So the point $YZ\cap AB$ is $\infty$, the only (projective) point on both $AB$ and $EE^*$. (Because $AB\cap EE^*=\infty_d=\infty$.) This implies $\infty\in YZ$, i.e. $YZ$ is parallel to the direction $(d)$.

Claim: The points $C,D,Z$ are colinear.

Proof of the claim: Let us denote by $x,y,t,u$ the arc lengths for the following (marked) arcs:

  • the arc from $A$ to the first intersection of $PO$ with $\overset\frown{AB}$
  • the arc the first intersection of $PO$ with $\overset\frown{AB}$ to $B$, so $x+y=\overset\frown{AB}$,
  • the arc $\overset\frown{BY}=\overset\frown{ZC}$,
  • the arc $\overset\frown{XZ}$.

Then the Theorem of Ceva in the triangle $\Delta ABO$ w.r.t. the inner point $E$ is: $$ \begin{aligned} 1 &= \frac {\sin\widehat{EAB}} {\sin\widehat{OAE}} \cdot \frac {\sin\widehat{AOE}} {\sin\widehat{EOB}} \cdot \frac {\sin\widehat{OBE}} {\sin\widehat{EBA}} \ , \\[2mm] &\qquad\text{ or using our notations} \\[2mm] 1 &= \frac {\sin(t/2)\ \cdot\ \sin(x)\ \cdot\ \sin((\pi-x-y-t-u)/2)} {\sin(\pi-x-y-t)/2\ \cdot\ \sin(y)\ \cdot\ \sin((t+u)/2)} \ . \end{aligned} $$ This is the point where we must use the important assumption on $E$, the fact that it is on $PO$.

We will show the colinearity of $C,D,Z$ by showing that $CD$ and $CZ$ is the same line. The angle $\widehat{ACB'}$ is fixed, so we take each of the lines, the sines of the angles separated by them from $\widehat{ACB'}$, and show equivalently the relation: $$ \tag{$*$} \frac {\color{gray}{AC\cdot} \sin\widehat{ACZ}} {\color{gray}{B'C\cdot} \sin\widehat{ZCB'}} \ \overset{(!)}{=\!\!=} \ \frac {\color{gray}{ AC\cdot CD\cdot} \sin\widehat{ACD}} {\color{gray}{B'C\cdot CD\cdot} \sin\widehat{DCB'}} \ . $$ Note that the R.H.S. above is the quotient of the areas of the triangles $\Delta ACD$ and $\Delta DCB'$, and we can write $$ RHS = \frac {\operatorname{Area}(ACD)} {\operatorname{Area}(DCB')} = \frac{AD}{DB'} = \frac {\operatorname{Area}(ABD)} {\operatorname{Area}(DBB')} = \frac {AB\cdot\sin\widehat{ABX}} {BB'\cdot\sin\widehat{XBB'}} \ . $$ The relation $(*)$ is thus equivalent to $$ \tag{$\dagger$} \frac {AC\cdot \sin(t/2)} {B'C\cdot \sin((\pi-x-y-t)/2)} \ \overset{(!)}{=\!\!=} \ \frac {AB \cdot\sin((t+u)/2)} {BB'\cdot\sin((\pi-x-y-t-u)/2)} \ . $$ Using the Ceva relation above, this is equivalent to $$ \tag{$\diamondsuit$} 1 \ \overset{(!)}{=\!\!=} \ \frac{AC}{B'C}\cdot \frac{BB'}{AB}\cdot \frac{\sin y}{\sin x}\ . $$ We can finally use the definition of $C$ as the point making the tangent in $A$, the line $EO$, and the line $BC$ concurent in $P$. We use for this the similarity $\Delta PAB\sim\Delta PCA$, which becomes $\displaystyle \frac{PC}{PA} = \frac{PA}{PB} = \frac{AC}{AB} $. So we replace in $(\diamondsuit)$ the fraction $AC:AB$ by $PA:PB$. We can finally show $(\diamond)$: $$ \frac{AC}{AB}\cdot \frac{B'C}{BC}\cdot \frac{\sin y}{\sin x} = \frac{PA}{PB}\cdot \frac{1}{\sin \widehat{B'BC}}\cdot \frac{PO\cdot OB\;\sin y}{PO\cdot OA\;\sin x} = \frac{PA}{PB}\cdot \frac{1}{\sin \widehat{B'BC}}\cdot \frac {\operatorname{Area}(POB)} {\operatorname{Area}(POA)} \\ = \frac{PA}{PB}\cdot \frac{1}{\sin \widehat{B'BC}}\cdot \frac {PB\cdot BO\;\sin\widehat{PBO}} {PA\cdot AO\;\sin\widehat{PAO}} = \frac {\sin \widehat{PBO}} {\sin \widehat{B'BC}} =1\ . $$

$\square$

This concludes the proof.


Corollary: This concludes the problem in the OP, since $$ \widehat{BAE} = \widehat{BAY} = \widehat{AYZ} = \widehat{ACZ} = \widehat{ACD} \ . $$

$\square$


Unfortunately i could not make the proof more structural. My hope was that it should be a bridge using results like Pascal's theorem and / or configurations of the shape Pappus, Desargue...

As bonus, here are some colinearities that can be established using Pascal's theorem.

Mathematics stackexchange problem 3107126, dan_fulea

The problem in the above tempting picture is that at the place where the point $D$ is placed, there are in fact three points,

  • the point $D:=AB'\cap BX$,
  • and the point $D_1:=CZ\cap BX$,
  • and the point $D_2:=CZ\cap AB'$.

And we need the equality of two of them.

To see colinearities from the picture, apply Pascal's theorem for instance for...

  • $AABXA'B'$, giving the colinearity of $D^*$, $\infty$, $D$, as mentioned above,
  • $XYZCAB$, giving the colinearity of $F:=XY\cap CA$, $\infty$, $D_1$,
  • $AA'XBB'Y$, giving the colinearity of $O$, $H:=A'X\cap B'Y$, $E$,
  • $BCAB'YXX$, giving the colinearity of $G:=BC\cap B'Y$, $F$, $D$,
  • $AYXA'CA$, giving the colinearity of $J:=AY\cap A'C$, $F$, $D^*$,
  • $AYZCBA$, ...
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